Method of determining the impedance of an electrochemical system

ABSTRACT

The invention relates to a method of determining the complex impedance Z(f m ) of a non-steady electrochemical system, comprising the following steps consisting in: bringing the system to a selected voltage condition and applying a sinusoidal signal with frequency f m  thereto; immediately thereafter, measuring successive values for voltage and current at regular time intervals ?T; calculating the discrete Fourier transforms for voltage (E(f)) and current (I(f)), the voltage transform being calculated for the single frequency f m  of the sinusoidal signal and the current transform being calculated for frequency f m  and for two adjacent frequencies f m−1  and f m+1  on either side of frequency f m ; and calculating the impedance using the following formula: Z(f m )=E(f m )/I*(f m ), wherein I* denotes a corrected current such that Re[I*(f m )]=Re[I(f m )]−{Re[I(f m+1 )]+Re[I(f m−1 )]}/2, Im[I*(f m )]=Im[I(f m )]−{Im[I(f m+1 )]+Im[I(f m−1 )]}/2.

To determine properties and qualities of an electrochemical system (suchas a cell, a battery, an electrodeposition system, a system foranalyzing a medium) and predict its future operation, one of the basicparameters is the system's impedance.

Generally, an electrochemical system comprises, in an electrolyticmedium, two main electrodes—a work electrode and a back-electrode. Areference electrode is arranged in the vicinity of the work electrodeand is used in relation therewith to perform various measurement orregulation operations. In the following, the mentioned voltages andcurrents correspond to measurements performed between a work electrodeand the corresponding reference electrode.

The impedance of an electrochemical system corresponds to thevoltage/current ratio. It is known that to rapidly measure the value ofthis impedance according to frequency, it is desirable to calculate theratio of the discrete Fourier transforms (DFT) of the voltage and of thecurrent. For this purpose, an excitation signal of small amplitude isapplied between a reference electrode and a work electrode of anelectrochemical cell, and N successive values e(n) and i(n) of thevoltage and of the current, with 1≦n≦N, are measured at equal timeintervals ΔT. The general expression of discrete Fourier transformsE(f_(m)) and I(f_(m)) of the voltage and of the current for a sequenceof N points is: $\begin{matrix}{{E\left( f_{m} \right)} = {\Delta\quad T{\sum\limits_{n = 1}^{N}\quad{{e(n)}\quad\exp\quad\left( {{- 2}\quad\pi\quad{jf}_{m}n\quad\Delta\quad T} \right)}}}} & (1) \\{{I\left( f_{m} \right)} = {\Delta\quad T{\sum\limits_{n = 1}^{N}\quad{{i(n)}\quad\exp\quad\left( {{- 2}\quad\pi\quad{jf}_{m}n\quad\Delta\quad T} \right)}}}} & (2)\end{matrix}$where j designates the complex number having −1 as a square, Ndesignates the number of measurement points, ΔT the sampling interval.The calculation of the discrete Fourier transform may be performed forN/2 values of frequency f_(m), with 0≦m<(N/2)−1, N being an even number.These N/2 discrete frequencies are regularly distributed between 0 and½ΔT (0 . . . 1/mΔT . . . ½ΔT).

The complex impedance for a given frequency f_(m) is then equal to:Z(f _(m))=E(f _(m))/I(f _(m)).  (3).

The impedance measurement methods used in practice, mainly differ by thetype of excitation signal of the system: sinusoidal, multi-sinusoidal,white noise, etc. A sinusoidal excitation is by far the most widelyused, since it appears to be the most accurate. The method consists ofimposing a sinusoidal regulation to the electrochemical cell with asignal of small amplitude and of recording the cell current and voltageresponse. The ratio of the Fourier transforms of the voltage and of thecurrent at the sinus frequency will give the impedance value at thisfrequency. The frequency spectrum is swept by modifying the excitationfrequency.

In the theory of electric systems, the expression of the impedance(equation (3)) is correct, provided that the analyzed system is linearand stationary and that disturbances are not introduced by phenomenaexternal to the system. In the case of electrochemical systems, thefulfilling of these conditions imposes specific precautions. Thetransfer function of an electrochemical system is generally non-linearbut it can be considered as linear over a small portion. This is why alow-amplitude excitation signal is used. Accordingly, thesignal-to-noise ratio decreases and the measurement time, that is, thenumber of measurement points, must be increased to suppress the noise byan integration of the response. However, the lengthening of themeasurement time sets problems regarding to the stationarity of theelectrochemical system. In many real cases, the stationarity conditioncannot be fulfilled. The causes are many: voltage relaxation, currentrelaxation, concentration relaxation. In such conditions, if themeasurement time is sufficiently long to have a good signal-to-noiseratio, the system is not stationary during the measurement time and thecalculated impedance has no great meaning any more, especially at lowfrequencies (for example, smaller than one hertz) where non-stationarityproblems are particularly serious.

Various methods have been provided to solve this problem of thenon-stationarity of electrochemical systems, but none has yielded asatisfactory solution.

Thus, an object of the present invention is to provide a novel methodfor calculating the impedance of an electrochemical system enablingignoring errors linked to the system non-stationarity.

To achieve this object, the present invention provides a method fordetermining the complex impedance Z(f_(m)) of a non-stationaryelectrochemical system, comprising the steps of:

setting the system to a selected voltage state and applying a sinusoidalsignal of frequency f_(m) thereto,

measuring, immediately after, successive values of the voltage and ofthe current at regular time intervals ΔT,

calculating the discrete Fourier transforms of the voltage (E(f)) and ofthe current (I(f)), the voltage transform being calculated for the solefrequency f_(m) of the sinusoidal signal and the current transform beingcalculated for frequency f_(m) and for two adjacent frequencies f_(m−1)and f_(m+1) on either side of frequency f_(m), and

calculating the impedance according to the following formula:Z(f _(m))=E(f _(m))/I*(f _(m))where I* designates a corrected current such that:Re[I*(f _(m))]=Re[I(f _(m))]−{Re[I(f _(m+1))]+Re[I(f _(m−1))])}/2Im[I*(f _(m))]=Im[I(f _(m))]−{Im[I(f _(m+1))]+Im[I(f _(m−1))]}/2.

A specific embodiment of the present invention will be non-limitinglydiscussed in relation with the appended drawing which shows theamplitude spectrum according to the frequency of an electrochemicalsystem submitted to a sinusoidal excitation of small amplitude and to avoltage step.

It should first be noted that, since the applied voltage is imposed bythe analysis tool, the system non-stationarity can only appear ascurrent variations.

The present invention is based on the analysis of the behavior of anelectrochemical system submitted to a voltage step. The DFT of thecurrent response mainly translates relaxation phenomena and thus theeffect of the system non-stationarity. As shown by the curve in dottedlines on the single drawing, the spectrum of amplitude I of the unit ofthe current DFT usually exhibits a strong low-frequency response tovoltage steps.

Conversely, the response to a sinusoidal excitation of frequency f_(m)of a stationary system translates as a single line at frequency f_(m).In practice, this response corresponds on the one hand to the systemresponse to the excitation at frequency f_(m), and on the other hand tothe contribution of relaxation effects.

The advantage of the use of the Fourier transform is that thesingle-frequency spectrum linked to the sinusoidal excitation issuperposed to the spectrum linked to the voltage step.

According to the present invention, once the above considerations havebeen taken into account, it is provided to subtract from the intensityresponse at frequency f_(m) the contribution due to relaxations,evaluated based on the analysis of the DFT at frequencies adjacent tof_(m). Indeed, the Fourier transform for frequencies f_(m−1) and f_(m+1)of the system will only correspond to non-stationarity phenomena and itwill be considered that the amplitude value linked to non-stationaritiesfor a frequency f_(m) is the average of the values for the two adjacentfrequencies.

Thus, the present invention provides:

setting an electrochemical system to a selected voltage state andapplying a single-frequency sinusoidal excitation thereto,

measuring, without waiting for the system stabilization, the amplitudeof the current and of the voltage at regular intervals immediately afterapplication of the voltage while the sinusoidal excitation is applied,and

calculating, on the one hand, the discrete Fourier transform of thevoltage for frequency f_(m) of the excitation and on the other hand theDFT of the current for value f_(m) and for two frequencies f_(m−1) andf_(m+1) adjacent to frequency f_(m), and

calculating the value of the complex impedance at frequency f_(m) basedon the value of the TFD of the voltage for frequency f_(m) and on acorrected value of the TFD of the current, taking into account the TFDscalculated for frequencies f_(m−1), f_(m), and f_(m+1).

Corrected amplitude I* of the discrete Fourier transform of the currentwill be calculated in real value and in imaginary value by the twofollowing equations:Re[I*(f _(m))]=Re[I(f _(m))]−{Re[I(f _(m+1))+Re[I(f _(m−1))]}/2Im[I*(f _(m))]=Im[I(f _(m))]−{Im[I(f _(m+1))+Im[I(f _(m−1))]}/2.

The value of the impedance corrected with the non-stationarity effectsthen is:Z(f _(m))=E(f _(m))/I*(f _(m))with I*(f_(m))=Re[I*(f_(m))]+jI_(m)[I*(f_(m))]

As soon as the reading of the points is performed for frequency f_(m), asinusoidal signal can be applied to a new frequency and a new readingcan be performed, and so on.

An advantage of the present method is that it makes a correct impedanceanalysis on non-stationary electrochemical systems possible, especiallyat very low frequencies. At the same time, a considerable time gain isobtained for systems which stabilize slowly since with the correctionaccording to the present invention, it is no longer necessary to waitfor the stabilization after powering-on of the system to start animpedance analysis.

1. A method for determining the complex impedance Z(f_(m)) of anon-stationary electrochemical system, characterized in that itcomprises the steps of: setting the system to a selected voltage stateand applying a sinusoidal signal of frequency f_(m) thereto, measuring,immediately after, successive values of the voltage and of the currentat regular time intervals ΔT, calculating the discrete Fouriertransforms of the voltage (E(f)) and of the current (I(f)), the voltagetransform being calculated for the sole frequency f_(m) of thesinusoidal signal and the current transform being calculated forfrequency f_(m) and for two adjacent frequencies f_(m−1) and f_(m+1) oneither side of frequency f_(m), and calculating the impedance accordingto the following formula:Z(f _(m))=E(f _(m))/I*(f _(m)) where I* designates a corrected currentsuch that:Re[I*(f _(m))]=Re[I(f _(m))]−{Re[I(f _(m+1))]+Re[I(f _(m−1))]}/2Im[I*(f _(m))]=Im[I(f _(m))]−{Im[I(f _(m+1))]+Im[I(f _(m−1))]}/2.
 2. Themethod of claim 1, characterized in that it is repeated for a successionof excitation frequencies.